PYTHAGORAS 141 asmuch as by the addition of successive gnomons consist- ing each of an odd number of unit squares to the original square unit or monad the square form was preserved. (19) In like manner, if the simplest oblong (erf/oo'^/ces), consist- ing of two unit squares or monads in juxtaposition, be taken and four unit squares be placed about it after the manner of a gnomon, and then in like manner six, eight . . . unit squares be placed in succession, the oblong form will be preserved. (20) Another of his doctrines was, that of all solid figures the sphere was the most beautiful, and of all plane figures the circle. 1 (21) According to lamblichus the Pythagoreans are said to have found the quadrature of the circle. 2 On examining the purely geometrical work of Pythagoras and his early disciples, as given in the preceding extracts, we observe that it is much concerned with the geometry of areas, and we are indeed struck with its Egyptian character. This appears in the theorem (3) concerning the filling up a plane with regular figures for floors or walls covered with tiles of various colours were common in Egypt ; in the construction of the regular solids (8), for some of them are found in Egyptian architecture ; in the problems con- cerning the application of areas (5) ; and lastly, in the theorem of Pythagoras (11), coupled with his rule for the construction of right- angled triangles in numbers (12). We learn from Plutarch that the Egyptians were acquainted with the geometrical fact that a triangle whose sides contain three, four, and five parts is right- angled, and that the square of the greatest side is equal to the squares of the sides containing the right angle. It is probable too that this theorem was known to them in the simple case where the right-angled triangle is isosceles, inasmuch as it would be at once suggested by the contemplation of a floor covered with square tiles the square on the diagonal and the sum of the squares on the sides contain each four of the right-angled triangles into which one of the squares is divided by its diagonal. It is easy now to see how the problem to construct a square which shall be equal to the sum of two squares could, in some cases, be solved numerically. From the observation of a chequered board it would be perceived that the element in the successive formation of squares is the gnomon or carpenter's square. Each gnomon consists of an odd number of squares, and the successive gnomons correspond to the successive odd numbers, and include, therefore, all odd squares. Suppose, now, two squares are given, one consisting of sixteen and the other of nine unit squares, and that it is proposed to form from them another square. It is evident that the square consisting of nine unit squares can take the form of the fourth gnomon, which, being placed round the former square, will generate a new square containing twenty-five unit squares. Similarly it may have been observed that the twelfth gnomon, consisting of twenty-five unit squares, could be transformed into a square each of whose sides contains five units, and thus it may have been seen conversely that the latter square, by taking the gnomonic or generating form with respect to the square on twelve units as base, would produce the square of thirteen units, and so on. This method required only to be generalized in order to enable Pythagoras to arrive at his rule for finding right-angled triangles whose sides can be expressed in numbers, which, we are told, sets out from the odd numbers. The nth square together with the nth gnomon forms the (?i + l)th square ; if the ?ith gnomon contains m 2 unit squares, m being an wi 2 1 odd number, we have 2?H-l = m 2 , .-, n= - -- , which gives the m rule of Pythagoras. The general proof of Euclid I. 47 is attributed to Pythagoras, but we have the express statement of Proclus (op. cit. , p. 426) that this theorem was not proved in the first instance as it is in the Elements. The following simple and natural way of arriving at the theorem is suggested by Bretschneider after Camerer. 8 A square can be dissected into the sum of two squares and two equal rectangles, as in Euclid II. 4 ; these two rectangles can, by draw- ing their diagonals, be decomposed into four equal right-angled triangles, the sum of the sides of each being equal to the side of the square ; again, these four right-angled triangles can be placed so that a vertex of each shall be in one of the corners of the square in such a way that a greater and less side are in continuation. gnomons are added successively in this manner to a square monad, the first gnomon may be regarded as that consisting of three square monads, and is indeed the constituent of a simple Greek fret ; the second of five square monads, &c. ; hence we have the gnomonic .lumbers. 1 Diog. Laert., De Vit. Pyth., viii. 19. Simplicius, In Aristotelis Pnysicorum libros quattuor priores Com- meniaria, ed. H. Diels, p. 60. _ 3 See Bretsch., Die Geom. vor Euklides, p. 82 ; Camerer, Eudidis Elem., vol. i. p. 444, and the references given there. The original square is thus dissected into the four triangles as before and the figure within, which is the square on the hypotenuse. This square, therefore, must be equal to the sum of the squares on the sides of the right-angled triangle. It is well known that the Pythagoreans were much occupied with the construction of regular polygons and solids, which in their cosmology played an essential part as the fundamental forms of the elements of the universe. We can trace the origin of these mathematical speculations in the theorem (3) that "the plane around a point is completely filled by six equilateral triangles, four squares, or three regular hexagons." Plato also makes the Pythagorean Timseus explain " Each straight-lined figure consists of triangles, but all triangles can be dissected into rectangular ones which are either isosceles or scalene. Among the latter the most beautiful is that out of the doubling of which an equilateral arises, or in which the square of the greater perpendicular is three times that of the smaller, or in which the smaller perpendicular is half the hypotenuse. But two or four right-angled isosceles triangles, properly put together, form the square ; two or six of the most beautiful scalene right-angled triangles form the equi- lateral triangle ; and out of these two figures arise the solids which correspond with the four elements of the real world, the tetra- hedron, octahedron, icosahedron, and the cube" 4 (Timseiis, 53, 54, 55). The construction of the regular solids is distinctly ascribed to Pythagoras himself by Eudemus (8). Of these five solids three the tetrahedron, the cube, and the octahedron were known to the Egyptians and are to be found in their architecture. Let us now examine what is required for the construction of the other two solids the icosahedron and the dodecahedron. In the formation of the tetrahedron three, and in that of the octahedron four, equal equilateral triangles had been placed with a common vertex and adjacent sides coincident ; and it was known that if six such triangles were placed round a common vertex with their adjacent sides coincident, they would lie in a plane, and that, therefore, no solid could be formed in that manner from them. It remained, then, to try whether five such equilateral triangles could be placed at a common vertex in like manner ; on trial it would be found that they could be so placed, and that their bases would form a regular pentagon. The existence of a regular pentagon would thus become known. It was also known from the formation of the cube that three squares could be placed in a similar way with a common vertex ; and that, further, if three equal and regular hexagons were placed round a point as common vertex with adjacent sides coincident, they would form a plane. It re- mained in this case too only to try whether three equal regular pentagons could be placed with a common vertex and in a similar way ; this on trial would be found possible and would lead to the construction of the regular dodecahedron, which was the regular solid last arrived at. We see that the construction of the regular pentagon is reqiiired for the formation of each of these two regular solids, and that, therefore, it must have been a discoveiy of Pythagoras. If we examine now what knowledge of geometry was required for the solution of this problem, we shall see that it depends on Euclid IV. 10, which is reduced to Euclid II. 11, which problem is reduced to the following : To produce a given straight line so that the rect- angle under the whole line thus produced and the produced part shall be equal to the square on the given line, or, in the language of the ancients, To apply to a given straight line a rectangle which shall be equal to a given area in this case the square on the given line and which shall be excessive by a square. Now it is to be observed that the problem is solved in this manner by Euclid (VI. 30, 1st method), and that we know on the authority of Eudemus that the problems concerning the application of areas and their excess and defect are old, and inventions of the Pythagoreans (5). Hence the statements of lamb'lichus concerning Hippasus (9) that he divulged the sphere with the twelve pentagons and of Lucian and the scholiast on Aristophanes (10) that the penta- gram was used as a symbol of recognition amongst the Pythagoreans become of greater importance. Further, the discovery of irrational magnitudes is ascribed to Pythagoras by Eudemus (13), and this discovery has been ever regarded as one of the greatest of antiquity. It is commonly assumed that Pythagoras was led to this theory from the considera- tion of the isosceles right-angled triangle. It seems to the present writer, however, more probable that the discovery of incommen- surable magnitudes was rather owing to the problem : To cut a line in extreme and mean ratio. From the solution of this problem it follows at once that, if on the greater segment of a line so cut a part be taken equal to the less, the greater segment, regarded as a new line, will be cut in a similar manner ; and this process can be continued without end. On the other hand, if a similar method be adopted in the case of any two lines which can be re- presented numerically, the process would end. Hence would arise 4 The dodecahedron was assigned to the fifth element, quinta pars, aether, or, as some think, to the universe. (See PHILOLADS.)
